Abstract:
The main goal of our work is to formally prove the correctness of the key
commands of the SCHUR software, an interactive program for calculating
with characters of Lie groups and symmetric functions. The core of the
computations relies on enumeration and manipulation of combinatorial
structures. As a first "proof of concept", we present a formal proof of the
conjugate function, written in C. This function computes the conjugate
of an integer partition. To formally prove this program, we use the Frama-C
software. It allows us to annotate C functions and to generate proof
obligations, which are proved using several automated theorem provers.
In this paper, we also draw on methodology, discussing on how to
formally prove this kind of program.

Abstract:
Decomposing the domain of a function into parts has many uses in mathematics.
A domain may naturally be a union of pieces, a function may be defined by
cases, or different boundary conditions may hold on different regions. For any
particular problem the domain can be given explicitly, but when dealing with a
family of problems given in terms of symbolic parameters, matters become more
difficult. This article shows how hybrid sets, that is multisets allowing
negative multiplicity, may be used to express symbolic domain decompositions
in an efficient, elegant and uniform way, simplifying both computation and
reasoning. We apply this theory to the arithmetic of piecewise functions and
symbolic matrices and show how certain operations may be reduced from
exponential to linear complexity.

Abstract:
We prove formally that the first order theory of algebraically closed
fields enjoys quantifier elimination, and hence is decidable. This proof
is organized in two modular parts. We first reify the first order
theory of rings and prove that quantifier elimination leads to
decidability. Then we implement an algorithm which constructs a
quantifier free formula from any first order formula in the theory of
ring. If the underlying ring is in fact an algebraically closed field,
we prove that the two formulas have the same semantic. The algorithm
producing the quantifier free formula is programmed in continuation
passing style, which leads to both a concise program and an elegant
proof of semantic correctness.

Abstract:
Computational content encoded into constructive type theory proofs can be
used to make computing experiments over concrete data structures.
In this paper, we explore this possibility when working in Coq with
chain complexes of infinite type (that is to say, generated by infinite sets)
as a part of the formalization of a hierarchy of homological algebra
structures.

Abstract:
We propose a formal study of interval analysis that concentrates on theoretical
aspects rather than on computational ones.
In particular we are interested in conditions for regularity of interval
matrices. An interval matrix is called regular if all scalar matrices included
in the interval matrix have non-null determinant and it is called singular
otherwise. Regularity plays a central role in solving systems of linear
interval equations. Several tests for regularity are available and widely used,
but sometimes rely on rather involved results, hence the interest in formally
verifying such conditions of regularity. In this paper we set the basis for
this work: we define intervals, interval matrices and operations on them in
the proof assistant Coq, and verify criteria for regularity and singularity of
interval matrices.

Reducing Expression Size using Rule-Based IntegrationAlbert Rich and David Jeffrey

Abstract:
This paper describes continuing progress on the development of a repository of
transformation rules
relevant to indefinite integration.
The methodology, however, is not restricted to integration.
Several optimization goals are being pursued, including achieving the best form
for the output,
reducing the size of the repository while retaining its scope,
and minimizing the number of
steps required for the evaluation process.
New optimizations for expression size are presented.

A Unified Formal Description of Arithmetic and Set Theoretical Data TypesPaul Tarau

Abstract:
We provide a ``shared
axiomatization'' of natural numbers and
hereditarily finite sets built around a polymorphic
abstraction of bijective base-2 arithmetics.
The ``axiomatization'' is described as a progressive refinement of Haskell type
classes with examples of instances converging to an efficient implementation in
terms of arbitrary length integers and bit operations.
As an instance, we derive algorithms to perform arithmetic
operations efficiently directly with hereditarily finite sets.
The self-contained source code of the paper is available at
http://logic.cse.unt.edu/tarau/research/2010/unified.hs.